2017
DOI: 10.1016/j.physa.2017.04.151
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Area-width scaling in generalised Motzkin paths

Abstract: We consider a generalised version of Motzkin paths, where horizontal steps have length $\ell$, with $\ell$ being a fixed positive integer. We first give the general functional equation for the area-length generating function of this model. Using a heuristic ansatz, we derive the area-length scaling behaviour in terms of a scaling function in one variable for the special cases of Dyck, (standard) Motzkin and Schr\"oder paths, before generalising our approach to arbitrary $\ell$. We then derive an expression for… Show more

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Cited by 1 publication
(2 citation statements)
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References 22 publications
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“…The special case w = 0 was analysed in [8]. In Table 1, we summarize the values of the critical exponents for w = − 1 9 and w > − 1 9 . Figure 4 shows a schematic picture of the phase diagram for fixed jump weight w ≥ − 1 9 .…”
Section: The Main Resultsmentioning
confidence: 99%
See 1 more Smart Citation
“…The special case w = 0 was analysed in [8]. In Table 1, we summarize the values of the critical exponents for w = − 1 9 and w > − 1 9 . Figure 4 shows a schematic picture of the phase diagram for fixed jump weight w ≥ − 1 9 .…”
Section: The Main Resultsmentioning
confidence: 99%
“…9 in [6]). By applying the method of steepest descents to the exact solution for the generating function, the same type of scaling behaviour has been proven rigorously to hold for staircase polygons and Dyck and Schröder paths [7,8,9]. These models can be interpreted as directed subclasses of SAP, with a lower sawtooth boundary in the latter two cases.…”
Section: Introductionmentioning
confidence: 85%